Answer :
Certainly! Let's solve the given system of linear equations step-by-step.
The system of equations is:
[tex]\[ \begin{array}{l} x + y = 11 \\ 5x - y = 10 \end{array} \][/tex]
### Step 1: Write the equations in standard form
Our given equations are already in standard form:
1. [tex]\(x + y = 11\)[/tex]
2. [tex]\(5x - y = 10\)[/tex]
### Step 2: Use the method of substitution or elimination
#### Using the Elimination Method:
1. Add the two equations to eliminate [tex]\(y\)[/tex]:
[tex]\[ (x + y) + (5x - y) = 11 + 10 \][/tex]
Simplifying this gives:
[tex]\[ x + y + 5x - y = 21 \][/tex]
[tex]\[ 6x = 21 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{21}{6} \][/tex]
[tex]\[ x = 3.5 \][/tex]
2. Substitute [tex]\(x = 3.5\)[/tex] back into the first equation to solve for [tex]\(y\)[/tex]:
[tex]\[ x + y = 11 \][/tex]
[tex]\[ 3.5 + y = 11 \][/tex]
Subtract 3.5 from both sides:
[tex]\[ y = 11 - 3.5 \][/tex]
[tex]\[ y = 7.5 \][/tex]
### Conclusion:
The solution to the system of equations is [tex]\(x = 3.5\)[/tex] and [tex]\(y = 7.5\)[/tex]. Thus, the point [tex]\((x, y) = (3.5, 7.5)\)[/tex] satisfies both equations.
So, the correct answer is:
[tex]\[ (3.5, 7.5) \][/tex]
The system of equations is:
[tex]\[ \begin{array}{l} x + y = 11 \\ 5x - y = 10 \end{array} \][/tex]
### Step 1: Write the equations in standard form
Our given equations are already in standard form:
1. [tex]\(x + y = 11\)[/tex]
2. [tex]\(5x - y = 10\)[/tex]
### Step 2: Use the method of substitution or elimination
#### Using the Elimination Method:
1. Add the two equations to eliminate [tex]\(y\)[/tex]:
[tex]\[ (x + y) + (5x - y) = 11 + 10 \][/tex]
Simplifying this gives:
[tex]\[ x + y + 5x - y = 21 \][/tex]
[tex]\[ 6x = 21 \][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{21}{6} \][/tex]
[tex]\[ x = 3.5 \][/tex]
2. Substitute [tex]\(x = 3.5\)[/tex] back into the first equation to solve for [tex]\(y\)[/tex]:
[tex]\[ x + y = 11 \][/tex]
[tex]\[ 3.5 + y = 11 \][/tex]
Subtract 3.5 from both sides:
[tex]\[ y = 11 - 3.5 \][/tex]
[tex]\[ y = 7.5 \][/tex]
### Conclusion:
The solution to the system of equations is [tex]\(x = 3.5\)[/tex] and [tex]\(y = 7.5\)[/tex]. Thus, the point [tex]\((x, y) = (3.5, 7.5)\)[/tex] satisfies both equations.
So, the correct answer is:
[tex]\[ (3.5, 7.5) \][/tex]